I have read this before and I have downloaded the BitCoin client, so how do I identify my private key here? I also noticed with my wallet, I only have one bitcoin address, how do I have so many more on CoinBase?
You can export your private key in the debug console, however, don't do that. It is intentionally made difficult to do so that people don't do it and compromise their security.
It's important to create a backup of your wallet on a regular basis in case something should happen to your computer. The backup will contain all of your private keys, addresses, and transactions. You can create the backup in Bitcoin Core from the "File" menu.
Ok, thanks for the info, but what is the private key?
The private key is a number that is mathematically associated with the public key in a way that makes it possible to re-generate the public key if you only have the private key. However, it is impossible with any currently known mathematics to calculate the private key if you only have the public key. The mathematical relationship makes it possible to use the private key to mathematically calculate a signature value on arbitrary data. Given the signature and the public key it is possible to verify that the signature and the public key are both associated with the same private key.
This signature process is used by bitcoin to validate the inputs to transactions, it provides a way for the spender to prove that they are authorized to spend those inputs. If the owner of the bitcoins loses their private keys they will no longer be able to generate valid signatures, and therefore will not be able to spend their bitcoins at all. If anyone else gains access to the private keys, they will be able to create valid signatures and therefore will be able to authorize the spending/stealing of funds.
I seen on here that there is a game where people are trying to crack bitcoin addresses to get rewards. How private can it be if people are cracking them?
There is no known way to calculate a properly generated private key from the public key or from a properly created signature.
However, if I know my own private key, it's possible for me to (accidentally or intentionally) leave clues to what the value might be. Using those clues others might be able to reduce the set of possible private keys to something small enough that they could search all the possibilities until they find the right value.
It is also possible for me to use a cryptographically insecure method of generating my private key. Others can then try to guess the method I used so they can re-create the same private key. If I want it to be a game, I can then leave clues about the method I chose for generating the key.